class: center, middle, inverse, title-slide .title[ # Class 3b: Review of concepts in Probability and Statistics ] .author[ ### Business Forecasting ] --- <style type="text/css"> .remark-slide-content { font-size: 20px; } </style> --- layout: false class: inverse, middle # Confidence Intervals --- ### Confidence Intervals - We calculated the mean price in our sample - How confident are we that our estimates close to the parameter's value? - Confidence intervals measure uncertainty around the estimate --- ### Confidence Intervals - Suppose that we calculated the confidence interval to be: `$$\{1086.64, 1404.22\}$$` -- - But where are these numbers coming from? -- 1. The sampling distribution of the sample mean tells us how likely we are to get a point estimate which is far away from the true mean -- 2. The confidence interval uses this property of the sampling distribution to tell us where the true mean might be -- - Let's go through these statements 1-by-1 --- ### Sampling distribution **Q: How likely is it that a sample mean is far away from the true mean?** - Consider a hypothetical sampling distribution of a sample mean - Reminder: `\(\bar{x} \sim \mathcal{N}(\mu, \frac{\sigma}{\sqrt{n}})\)` -- - If we draw samples repeatedly, 95% of their means will be within the shaded area - Why 1.96? <img src="data:image/png;base64,#C_3_slides_b_files/figure-html/unnamed-chunk-1-1.png" width="100%" /> --- ### Sampling distribution - Assume we have `\(n>30\)` - CLT applies and `\(\bar{X} \sim N(\mu_X, \frac{\sigma_X}{\sqrt n})\)` -- - We want to find `\(k_1\)` and `\(k_2\)`, such that: - `\(P(k_1<\bar{X}<k_2)=0.95\)`, so -- - `\(P(\bar{X}<k_1)=0.025\)` and `\(P(\bar{X}>k_2)=0.025\)` (or `\(P(\bar{X}<k_2)=0.975\)`) -- - Trick is to standardize the variable: `$$P(\bar{X}<k_2)=P(\bar{X}-\mu_X<k_2-\mu_X)=P(\underbrace{\frac{\bar{X}-\mu_X}{ \frac{\sigma_X}{\sqrt n}}}_{Z \sim N(0,1)}<\underbrace{\frac{k_2-\mu_X}{ \frac{\sigma_X}{\sqrt n}}}_{k_2'})$$` -- In probability tables, we can find `\(k_2'\)`, such that `\(P(Z<k_2')=0.975\)` --- <img src="data:image/png;base64,#C_3_slides_b_files/figure-html/unnamed-chunk-2-1.png" width="100%" /> - It's 97.5% quantile of standard normal: `\(k_2'=v_{0.975}^Z=1.96\)`. `\(P(Z<v_{0.975})=P(Z<1.96)=0.975\)` -- - Let's go back to `\(k_2'=\frac{k_2-\mu_X}{ \frac{\sigma_X}{\sqrt n}}\)`, from which we can back-out `\(k_2\)` -- - `\(k_2=\mu_X+k_2'\frac{\sigma_X}{\sqrt n} =\mu_X+1.96\frac{\sigma_X}{\sqrt n}\)` -- - By symmetry of normal, `\(k'_1=-k'_2\)`, so `\(k_1=\mu_X+\frac{\sigma_X}{\sqrt n} k_1'=\mu_X-1.96\frac{\sigma_X}{\sqrt n}\)` --- <img src="data:image/png;base64,#C_3_slides_b_files/figure-html/unnamed-chunk-3-1.png" width="100%" /> - Another way to see it: - Our variable is a linear transformation of a standard normal: `\(\bar{X}=\mu_X+\frac{\sigma_X}{ \sqrt n} Z\)` -- - So its percentiles are linear transformation of st.nor. quantiles: `\(v_{0.975}^\bar{X}=\mu_X+\frac{\sigma_X}{\sqrt n}v_{0.975}^Z\)` --- ### Yet another way to see it `\begin{align*} 0.95&= P(-1.96<Z<1.96) \\ &= P(z_{-\frac{\alpha}{2}}<Z<z_{\frac{\alpha}{2}}) \\ &= P(z_{-\frac{\alpha}{2}}<\frac{\bar{X}-\mu}{\sigma/\sqrt n}<z_{\frac{\alpha}{2}}) \\ &= P(z_{-\frac{\alpha}{2}}\sigma/ \sqrt n<\bar{X}-\mu<z_{\frac{\alpha}{2}}\sigma/\sqrt n) \\ &= P(\mu-z_{-\frac{\alpha}{2}}\sigma/\sqrt n< \bar{X} <\mu+z_{\frac{\alpha}{2}}\sigma/\sqrt n) \\ \end{align*}` - Theoretically, CLT theorem guarantees that `\(\frac{\bar{X}-\mu}{\sigma/\sqrt n}\)` is standard normal -- - What happens if you do not know `\(\sigma\)`? -- - In large sample, `\(s \rightarrow \sigma\)`, so `\(\frac{\bar{X}-\mu}{s/\sqrt n} \rightarrow N(0,1)\)` - So in large samples, standardized sample mean (with estimated standard deviation) will also have normal distribution - You man need a bit higher n to ensure `\(s \rightarrow \sigma\)` --- ### Sampling distribution **Q: How far is the sampled mean from the true mean?** - Hence 95% of the draws of sample means will be within distance of `\(1.96\frac{\sigma_X}{\sqrt n}\)` to the true parameter - There is only 5% chance that we have draw sample weird enough that `\(\bar{X}\)` is further from `\(\mu_X\)` by more than `\(1.96\frac{\sigma_X}{ \sqrt n}\)` - Confidence interval of `\(\bar{X}\)` will cover `\(\mu_X\)` as long as `\(|\mu_X-\bar{X}|<1.96\frac{\sigma_X}{\sqrt n}\)` <img src="data:image/png;base64,#C_3_slides_b_files/figure-html/unnamed-chunk-4-1.png" width="100%" /> --- ### Sampling distribution - Suppose we draw many samples from the same distribution - For each sample we compute the sample mean and we construct the interval - 95% of them will cover the true population mean! <iframe src="https://seeing-theory.brown.edu/frequentist-inference/index.html#section2" width="100%" height="400px" data-external="1"></iframe> Source: [https://seeing-theory.brown.edu/frequentist-inference/index.html#section2) --- ### Calculation Procedure Use this procedure if `\(n>30\)` 1. Take an IID sample -- 2. Calculate mean `\(\bar{x}\)` and standard deviation `\(s\)` in your sample - Standard Error is standard deviation of the estimator `\(\small SE=\frac{s}{\sqrt n}\)` -- 3. Pick confidence level (usually 90,95,99%) - We typically denote the confidence level `\(1-\alpha\)` - `\(\alpha\)` is probability of making a Type 1 error (more about it later) - .blue[Example]: if confidence level is 95%, `\(\small \alpha=0.05\)` -- 4. Find the corresponding critical values `\(\small z_{\frac{\alpha}{2}}\)` - Critical values are such that `\(\small P(-z_{\frac{\alpha}{2}}<Z<z_{\frac{\alpha}{2}})=1-\alpha\)` - .blue[Example]: if confidence level is 95%, `\(\small z_\frac{\alpha}{2}=z_{0.025}=1.96\)` -- 5. Construct the confidence interval as: `$$\small \{\bar{x}- z_{\frac{\alpha}{2}}*\underbrace{\frac{s}{\sqrt n}}_{SE}, \bar{x}+ z_{\frac{\alpha}{2}}*\frac{s}{\sqrt n}\}$$` --- ### Finding Critical Values - Suppose confidence interval is 99%. - Then `\(\alpha=0.01\)` - We are looking for `\(z_{\frac{\alpha}{2}}\)` such that: `$$P(-z_{\frac{\alpha}{2}}<Z<z_{\frac{\alpha}{2}})=0.99$$` <img src="data:image/png;base64,#C_3_slides_b_files/figure-html/unnamed-chunk-5-2.png" width="100%" /> -- `$$P(Z>z_{0.005})=0.005 \qquad\text{or}\qquad P(Z<z_{0.005})=0.995$$` --- <iframe src="https://www.mathsisfun.com/data/standard-normal-distribution-table.html" width="100%" height="480px" data-external="1"></iframe> Source: [https://www.mathsisfun.com/data/standard-normal-distribution-table.html) --- ### Finding Critical Values `\(P(Z<z_{\frac{\alpha}{2}})=0.995\)` `\(z_{\frac{\alpha}{2}}\)`, is 99.5% quantile of standard normal `\(\rightarrow\)` `\(z_{\frac{\alpha}{2}}=2.58\)` <img src="data:image/png;base64,#C_3_slides_b_files/figure-html/unnamed-chunk-6-2.png" width="100%" /> <center> <img src=prob_table_normal.png width="800"> </center> --- ### Constructing CI: example Let's calculate 90% CI for average price of listing with grade>4.5 1. Take an IID sample - `\(n=100\)` `\(\checkmark\)` -- 2. Calculate mean `\(\bar{x}\)` and standard deviation `\(s\)` - `\(\bar{x}=\)` 1245.43 and `\(s=\)` 961.9 -- 3. Pick confidence level - We pick 90%, so `\(\alpha=0.1\)` -- 4. Find the corresponding critical values `\(z_{\frac{\alpha}{2}}\)` - Find `\(z_\frac{\alpha}{2}\)` such that `\(P(Z>z_{\frac{\alpha}{2}})=0.05\)` (or `\(P(Z<z_{\frac{\alpha}{2}})=0.95\)`) - `\(z_{0.05}=1.65\)` -- 5. Construct the confidence interval as: `$$\small \{\bar{x}- z_{\frac{\alpha}{2}}*\frac{s}{\sqrt n}, \bar{x}+ z_{\frac{\alpha}{2}}*\frac{s}{\sqrt n}\}$$` `$$\small \{1245.43- 1.65\frac{961.9}{\sqrt{100}}, 1245.43+ 1.65\frac{961.9}{\sqrt 100}\}$$` --- ### Interpreting confidence intervals `$$\small CI_{90}=\{1086.64, 1404.22\}$$` How do we interpret a 90% confidence interval we computed? - **Correct Interpretation** - We are 90% confident that the interval captures the true mean - We are 90% confident that the true mean price of listings with grade>4.5 is between 1086.64 and 1404.22 -- - **Incorrect** - With 90% probability the true mean is between 1086.64 and 1404.22 - Computed interval is not-random and true mean is not random, so can't make probabilistic statements. - Interval is a function of random variables only **before** we draw a sample and make any computation. - After we have a sample, nothing is random. The true mean is either between 1086.64 and 1404.22 or not. --- ### Shape of confidence intervals Confidence intervals `\(\small \{\bar{x}- z_{\frac{\alpha}{2}}*\frac{s}{\sqrt n}, \bar{x}+ z_{\frac{\alpha}{2}}*\frac{s}{\sqrt n}\}\)` are wider when: - Confidence level is higher (99% is wider than 90%) - When `\(n\)` is small - When `\(\sigma\)` is large <iframe src="https://seeing-theory.brown.edu/frequentist-inference/index.html#section2" width="100%" height="420px" data-external="1"></iframe> --- ### Practice Suppose we want to know what is the average commute time for ITAM students. We take a sample of 60 students. We calculate sample mean to be `\(\bar{x}=23\)` and sample standard deviation to be `\(s=8\)`. Calculate 99% confidence interval and say whether interpratation is correct: -- - We are 99% confident that the average commute of these 60 students is between (20.3364, 25.6636) -- - .red[False] -- - We are 99% confident that the average commute of all ITAM students is between (20.3364, 25.6636) -- - .green[True] -- - A 95% cofidence interval would be wider -- - .red[False] -- - 99% of random samples would have mean between (20.3364, 25.6636) -- - .red[False] -- - 99% of random samples would capture the true mean -- - .green[True] -- - With 99% probability true mean is between this and this (20.3364, 25.6636) -- - .red[False] --- ### What critical values? When should we use critical values from Normal Distribution? 1. Original distribution (of `\(X\)`) is not normal: - If `\(n>30\)` - you can use critical values from normal distribution - If `\(n<30\)` - you are screwed -- 2. Original distribution (of `\(X\)`) is normal: - If you know `\(\sigma\)`, you can use critical values from normal ( `\(n\)` doesn't matter) - If `\(X\)` is normal, then use `\(\sigma\)` instead of `\(s\)` and `\(\frac{\bar{X}-\mu}{\frac{\sigma}{\sqrt n}} \sim N(0,1)\)` - If you don't know `\(\sigma\)` but `\(n>30\)`, you can use critical values from normal - CLT kicks in - If you don't know `\(\sigma\)` and `\(n<30\)`, you use critical values from .blue[student's t]. - `\(\frac{\bar{X}-\mu}{\frac{s}{\sqrt n}}\)` is not normal. `\(s\)` is not a good approx. of `\(\sigma\)` when `\(n\)` is low --- ### What's Student's t? If `\(X_1\)`, `\(X_2\)`, . . . , `\(X_n\)` are i.i.d. from `\(N(µ, σ)\)`, then `$$T =\frac{\bar{X} − µ}{s/\sqrt n}$$` Where `\(s\)` is sample standard deviation. T has a student's t distribution with n−1 degrees of freedom `$$T \sim t_{n-1}$$` --- ### What's Student's t? - Bell shaped and symmetric around 0 - More spread out - heavier tails, more uncertainty (because we don't know standard deviation) - Shape determined by the degrees of freedom. - As n increases (and hence degrees of freedom), it tends to standard normal (as it should by CLT!) - Less uncertainty because we are better at estimating standard deviation <div id="htmlwidget-21248f5511e862f60cbf" style="width:800px;height:300px;" class="plotly html-widget"></div> <script type="application/json" 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(y) ","x"]},"highlight":{"on":"plotly_click","persistent":false,"dynamic":false,"selectize":false,"opacityDim":0.20000000000000001,"selected":{"opacity":1},"debounce":0},"shinyEvents":["plotly_hover","plotly_click","plotly_selected","plotly_relayout","plotly_brushed","plotly_brushing","plotly_clickannotation","plotly_doubleclick","plotly_deselect","plotly_afterplot","plotly_sunburstclick"],"base_url":"https://plot.ly"},"evals":[],"jsHooks":[]}</script> --- ### Student's t critical values Finding critical values for student's t distribution: 1. Determine what is the right number of degrees of freedom ( `\(n-1\)` )! 2. Determine what's your confidence level and your `\((1-\alpha)\)` - From this figure out `\(\alpha/2\)` 3. Find the percentile such that `$$P(T>t_{\frac{\alpha}{2},\underbrace{n-1}_{d.f.}})=\frac{\alpha}{2} \qquad\text{or}\qquad P(T<t_{\frac{\alpha}{2},n-1})=1-\frac{\alpha}{2}$$` <img src="data:image/png;base64,#C_3_slides_b_files/figure-html/unnamed-chunk-7-1.png" width="100%" /> --- ### Example - `\(n=10 \rightarrow df=9\)` - Confidence level is 95% `\(\rightarrow1-\alpha=0.95\)` and `\(\frac{\alpha}{2}=0.025\)` - What's `\(t_{0.025,9}\)` such that `\(P(T<t_{0.025,9})=0.975\)` <center> <img src=t_student_table.png width="1000"> </center> --- - `\(n=10 \rightarrow df=9\)` - Confidence level is 95% `\(\rightarrow1-\alpha=0.95\)` and `\(\frac{\alpha}{2}=0.025\)` - What's `\(t_{0.025,9}\)` such that `\(P(T<t_{0.025,9})=0.975\)` <img src="data:image/png;base64,#C_3_slides_b_files/figure-html/unnamed-chunk-8-1.png" width="100%" /> - Once we have critical value, we construct the CI as before: `$$\small \{\bar{x}- t_{\frac{\alpha}{2},n-1}*\frac{s}{\sqrt n}, \bar{x}+ t_{\frac{\alpha}{2},n-1}*\frac{s}{\sqrt n}\}$$` --- ### Practice: Your company implemented free shipping for a random group of customers. They want to know whether it increased spending. Here is your data: $157.80, $192.45, $210.20, $175.60, $198.30, $180.90, $205.75, $185.20, $177.40, $195.60 a) Calculate 90% confidence interval. What assumptions you need? -- - Hint 1: `\(\sum_ix_i=1869.80\)` - Hint 2: `\(\sum_ix_i^2=361752.55\)` -- b) Average spending without free shipping is $182, can say anything about whether free shipping increased spending? --- ### Confidence Intervals for Variance - Suppose `\(X_1, X_2, ...X_n\)` come from normal distribution - The sampling distribution of the sample variance is `\(S^2=\frac{\sum_i(X_i-\bar{X})^2}{n-1}\)` is: `$$\small \frac{(n-1)S^2}{\sigma^2}\sim\chi_{n-1}$$` -- - We will use the fact that: `$$\small P(\chi_{0.025,n-1}<\frac{(n-1)S^2}{\sigma^2}<\chi_{0.975,n-1})=0.95$$` <img src="data:image/png;base64,#C_3_slides_b_files/figure-html/unnamed-chunk-9-1.png" width="100%" /> --- ### Confidence Intervals for Variance How we use it to construct the confidence interval? `\begin{align*} 0.95&= P(\chi_{0.025,n-1}<\frac{(n-1)S^2}{\sigma^2}<\chi_{0.975,n-1}) \\ &= P(\frac{1}{\chi_{0.975,n-1}} < \frac{\sigma^2}{(n-1)S^2} < \frac{1}{\chi_{0.025,n-1}}) \\ &= P(\frac{(n-1)S^2}{\chi_{0.975,n-1}} < \sigma^2 < \frac{(n-1)S^2}{\chi_{0.025,n-1}}) \end{align*}` -- So more generally, the confidence interval for the sample variance is `$$CI_{1-\alpha}=\{\frac{(n-1)S^2}{\chi_{1-\frac{\alpha}{2},n-1}}, \frac{(n-1)S^2}{\chi_{\frac{\alpha}{2},n-1}}\}$$` - Where `\(\chi_{1-\frac{\alpha}{2},n-1}\)` and `\(\chi_{\frac{\alpha}{2},n-1}\)` are quantiles of `\(\chi_{n-1}\)` distribution, such that `\(P(X<\chi_{1-\frac{\alpha}{2},n-1})=1-\frac{\alpha}{2}\)` and `\(P(X<\chi_{\frac{\alpha}{2},n-1})=\frac{\alpha}{2}\)` - You can read them off the tables --- ### Practice Suppose you produce sausages. As a quality control, you measure the level of fat in your sausages. You take a random sample of 12 sausages and you find the variance of 20 ( `\(grams^2\)` ). Find 99% confidence interval for the variance. What assumptions you need?